If A and B are square matrices of the same order such that AB = BA, then show that (A + B)2 = A2 + 2AB + B2.
Given that A and B are square matrices of the same order such that AB = BA.
We need to prove that (A + B)2 = A2 + 2AB + B2.
We know (A + B)2 = (A + B)(A + B)
⇒ (A + B)2 = A(A + B) + B(A + B)
⇒ (A + B)2 = A2 + AB + BA + B2
However, here it is mentioned that AB = BA.
⇒ (A + B)2 = A2 + AB + AB + B2
∴ (A + B)2 = A2 + 2AB + B2
Thus, (A + B)2 = A2 + 2AB + B2 when AB = BA.